In this paper, we will consider blowup solutions to the so called Keller-Segel system and its simplified form. The Keller-Segel system was introduced to describe how cellular slime molds aggregate, owing to the motion of the cells toward a higher concentration of a chemical substance produced by themselves. We will describe a common conjecture in connection with blowup solutions to the Keller-Segel system, and some results for solutions to simplified versions of the Keller-Segel system, giving the...

A class of nonlinear viscous transport equations describing aggregation phenomena in biology is considered. General conditions on an interaction potential are obtained which lead either to the existence or to the nonexistence of global-in-time solutions.

Since the release of the Bio 2010 report in 2003, significant emphasis has been placed on initiating changes in the way undergraduate biology and mathematics courses are taught and on creating new educational materials to facilitate those changes. Quantitative approaches, including mathematical models, are now considered critical for the education of the next generation of biologists. In response, mathematics departments across the country have initiated changes to their introductory calculus sequence,...

We review the optimal design of an arterial bypass graft following either a (i) boundary optimal control approach, or a (ii) shape optimization formulation. The main focus is quantifying and treating the uncertainty in the residual flow when the hosting artery is not completely occluded, for which the worst-case in terms of recirculation effects is inferred to correspond to a strong orifice flow through near-complete occlusion.A worst-case optimal control approach is applied to the steady Navier-Stokes...

This paper is concerned with the two-species chemotaxis-Navier–Stokes system with Lotka–Volterra competitive kinetics $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\left(1\right)}_t+u·1=𝔻1-{\chi }_1·\left(1c\right)+{\mu }_{1}1(1-1-a_{1}2)\hfill & \text{in}\times (0,\infty ),\hfill \\ {\left(2\right)}_t+u·2=𝔻2-{\chi }_2·\left(2c\right)+{\mu }_{2}2(1-a_{2}1-2)\hfill & \text{in}\times (0,\infty ),\hfill \\ c_t+u·c=𝔻c-(\alpha 1+\beta 2)c\hfill & \text{in}\times (0,\infty ),\hfill \\ u_t+(u·)u=𝔻u+\nabla P+(\gamma 1+2)\Phi ,·u=0\hfill & \text{in}\times (0,\infty )\hfill \end{array}\right.\end{array}$$ under homogeneous Neumann boundary conditions and initial conditions, where is a bounded domain in R3 with smooth boundary. Recently, in the 2-dimensional setting, global existence and stabilization of classical solutions to the above system were first established. However, the 3-dimensional case has not been studied: Because of difficulties in the Navier–Stokes system, we can...

We study a quasilinear parabolic-parabolic chemotaxis system with nonlinear logistic source, under homogeneous Neumann boundary conditions in a smooth bounded domain. By establishing proper a priori estimates we prove that, with both the diffusion function and the chemotaxis sensitivity function being positive, the corresponding initial boundary value problem admits a unique global classical solution which is uniformly bounded. The result of this paper is a generalization of that of Cao (2014).

This paper deals with parabolic-elliptic chemotaxis systems with the sensitivity function $\chi \left(v\right)$ and the growth term $f\left(u\right)$ under homogeneous Neumann boundary conditions in a smooth bounded domain. Here it is assumed that $0<\chi \left(v\right)\le {\chi }_0/v^k$$(k\ge 1...$

A growing body of literature testifies to the importance of quantitative reasoning skills in the 21st-century biology curriculum, and to the learning benefits associated with active pedagogies. The process of modeling a biological system provides an approach that integrates mathematical skills and higher-order thinking with existing course content knowledge. We describe a general strategy for teaching model-building in an introductory biology course,...

An increasing number of known RNA 3D structures contributes to the recognition of various RNA families and identification of their features. These tasks are based on an analysis of RNA conformations conducted at different levels of detail. On the other hand, the knowledge of native nucleotide conformations is crucial for structure prediction and understanding of RNA folding. However, this knowledge is stored in structural databases in a rather distributed form. Therefore, only automated methods...